Methods to solve this type of equations

Question

Is there any method to solve (find the number of solutions or in some problems find the solutions) problems of following type:

1.${}\quad x=\log_e x^2-1$
2.${}\quad \tan x+2x^2-3=0$

other than by separating polynomial function and other types of functions (like exponential, logarithmic) writing them on two sides of equality and drawing their graphs and noting where the two curves intersect?

Any sort of approach or method is welcome as long as it leads to an elegant solution and as is elementary.

Answer 1

Derivative and computing your functions at some chosen points

Continuity can also prove useful (intermediate value theorem)

Answer 2

There aren't any general methods. Not even for polynomials (there are formulas for the roots up to 4th degree, but there are 5th degree polynomials whose roots can't be expressed algebraically). Some special form polynomials can be reduced, there is the rational root test. You can use $\gcd(p(x), p'(x))$ to separate repeat roots, it is easy to show that if a polynomial $p$ has repeat roots, they are also roots of $p'$, so you get smaller degree polynomials with no repeat roots, easier to handle.

Specifically for polynomials there are numerical methods that converge to the roots.

There are plenty of numerical methods, but they require some guess at the root you are looking for. And the conditions under which they do converge aren't easy to verify in practice. Best bet is to try and see. Or use some of the less efficient algorithms that guarantee convergence when started with an interval straddling a root.

Answer 3

For the logarithmic=polynomial case, you can see if you have any luck reducing it to a form the Lambert W function can handle; I think that should work out for your first equation.

Of course, the Lambert function is an ad-hoc solution introduced to be able to solve equations which were previously 'unsolvable' and you might argue that this is not a general method. On the other hand, the W function is a perfectly normal one (like $\arcsin$, say) that can be studied with all the tools of analysis. So an approach for when you really get stuck would be to introduce something like the Lambert function, examine its properties and see if you get more insight.